# Inequities in the Shanks–Rényi prime number race: An asymptotic formula for the densities

• Daniel Fiorilli and Greg Martin

## Abstract

Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if a is a nonsquare (mod q) and b is a square (mod q), then there tend to be more primes congruent to a (mod q) than b (mod q) in initial intervals of the positive integers; more succinctly, there is a tendency for π (x; q, a) to exceed π (x; q, b). Rubinstein and Sarnak defined δ (q; a, b) to be the logarithmic density of the set of positive real numbers x for which this inequality holds; intuitively, δ (q; a, b) is the “probability” that π (x; q, a) > π (x; q, b) when x is “chosen randomly”. In this paper, we establish an asymptotic series for δ (q; a, b) that can be instantiated with an error term smaller than any negative power of q. This asymptotic formula is written in terms of a variance V (q; a, b) that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet L-functions corresponding to characters (mod q); we show how V (q; a, b) can be evaluated exactly as a finite expression. In addition to providing the exact rate at which δ (q; a, b) converges to 1/2 as q grows, these evaluations allow us to compare the various density values δ (q; a, b) as a and b vary modulo q; by analyzing the resulting formulas, we can explain and predict which of these densities will be larger or smaller, based on arithmetic properties of the residue classes a and b (mod q). For example, we show that if a is a prime power and a′ is not, then δ (q; a, 1) < δ (q; a′, 1) for all but finitely many moduli q for which both a and a′ are nonsquares. Finally, we establish rigorous numerical bounds for these densities δ (q; a, b) and report on extensive calculations of them, including for example the determination of all 117 density values that exceed 9/10.